Tools

"... Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set ..."

Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set of natural properties for discrete Laplace operators for triangular surface meshes. We prove an important theoretical limitation: discrete Laplacians cannot satisfy all natural properties; retroactively, this explains the diversity of existing discrete Laplace operators. Finally, we present a family of operators that includes and extends well-known and widely-used operators.

"... Abstract. In this paper, we apply Nyström method, a sub-sampling and reconstruction technique, to speed up spectral mesh processing. We first relate this method to Kernel Principal Component Analysis (KPCA). This enables us to derive a novel measure in the form of a matrix trace, based soly on sampl ..."

Abstract. In this paper, we apply Nyström method, a sub-sampling and reconstruction technique, to speed up spectral mesh processing. We first relate this method to Kernel Principal Component Analysis (KPCA). This enables us to derive a novel measure in the form of a matrix trace, based soly on sampled data, to quantify the quality of Nyström approximation. The measure is efficient to compute, well-grounded in the context of KPCA, and leads directly to a greedy sampling scheme via trace maximization. On the other hand, analyses show that it also motivates the use of the max-min farthest point sampling, which is a more efficient alternative. We demonstrate the effectiveness of Nyström method with farthest point sampling, compared with random sampling, using two applications: mesh segmentation and mesh correspondence. 1

"... We present a family of discrete isometric bending models (IBMs) for triangulated surfaces in 3-space. These models are derived from an axiomatic treatment of discrete Laplace operators, using these operators to obtain linear models for discrete mean curvature from which bending energies are assemble ..."

We present a family of discrete isometric bending models (IBMs) for triangulated surfaces in 3-space. These models are derived from an axiomatic treatment of discrete Laplace operators, using these operators to obtain linear models for discrete mean curvature from which bending energies are assembled. Under the assumption of isometric surface deformations we show that these energies are quadratic in surface positions. The corresponding linear energy gradients and constant energy Hessians constitute an efficient model for computing bending forces and their derivatives, enabling fast time-integration of cloth dynamics with a two- to three-fold net speedup over existing nonlinear methods, and near-interactive rates for Willmore smoothing of large meshes. Key words: cloth simulation, thin plates, Willmore flow, bending energy, discrete Laplace operator, discrete mean curvature, non-conforming finite elements. 1